Events

Faisal Romshoo, Department of Pure Mathematics, University of Waterloo

"A theoretical framework for H-structures"

For an oriented Riemannian manifold $(M^n, g)$, and Lie subgroup $H \subset SO(n)$, a compatible $H$-structure on $(M^n,g)$ is a principal $H$-subbundle of the principal $SO(n)$-bundle of oriented orthonormal coframes.  They can be described in terms of the sections of the homogeneous fibre bundle obtained by $H$-reduction of the oriented frame bundle. Examples of these structures include $U(m)$-structures, $G_2$-structures and $\text{Spin(7)}$-structures. In this talk, we will study a general theory for $H$-structures described in a paper of Daniel Fadel, Eric Loubeau, Andrés J. Moreno and Henrique N. Sá Earp titled "Flows of geometric structures" (https://arxiv.org/abs/2211.05197).

MC 5403

AJ Fong, Department of Pure Mathematics, University of Waterloo

"Flat families of schemes"

We will discuss families of schemes, and how flatness is the correct relevant notion for families.

MC 5417

Rahim Moosa, Department of Pure Mathematics, University of Waterloo

"Binding groups for rational dynamics"

I will report on ongoing work with Moshe Kamensky toward developing a theory of binding groups for quantifier-free types in ACFA, well-suited for applications to rational algebraic dynamics.

MC 5479

Talk #1: Ted Fu, University of Waterloo

"On Waring's problem for large powers"

Let G(k) be the least number s having the property that every sufficiently large natural number is the sum of at most s positive integer k-th powers. In this talk, I will present how Brüdern and Wooley implement smooth numbers technologies in their minor arc analysis and derive G(k) ≤ ⌈k(log k + 4.20032)⌉.

Talk #2: Aidan Boyle, University of Waterloo

"Waring’s problem: Beyond Freiman’s Theorem"

Suppose that we are given a non-decreasing sequence of positive integers (k_i) where each term is at least 2. Given a positive integer j, we seek to understand the circumstances in which there exists a positive integer s := s(j) such that every sufficiently large natural number n can be written as a sum of s positive integers to the respective powers k_j, ..., k_j+s-1. Freĭman asserted that such representation exists if and only if the infinite summation of all 1/k_i diverges. We provide an effective version of this theorem, and in particular, comment on instances in which the exponents form a sequence of consecutive terms of an arithmetic progression.

MC 5417

Alex Waldron, University of Wisconsin-Madison

"Parabolic gap theorems for the Yang-Mills functional"

Given a principal bundle over a compact Riemannian 4-manifold or special-holonomy manifold, it is natural to ask whether a uniform gap exists between the instanton energy and that of any non-minimal Yang-Mills connection. This question is quite open in general, although positive results exist in the literature. We'll review several of these gap theorems and strengthen them to statements of the following type: the space of all connections below a certain energy deformation retracts (under Yang-Mills flow) onto the space of instantons. As applications, we recover a theorem of Taubes on path-connectedness of instanton moduli spaces on the 4-sphere, and obtain a method to construct instantons on quaternion-Kähler manifolds with positive scalar curvature.

MC 5417

Katarzyna Wyczesany, Carnegie Mellon University

"Dualities on sets and how they appear in optimal transport"

In this talk, we will discuss order reversing quasi involutions, which are dualities on their image, and their properties. We prove that any order reversing quasi-involution is of a special form, which arose from the consideration of optimal transport problem with respect to costs that attain infinite values. We will discuss how this unified point of view on order reversing quasi-involutions helps to deepen the understanding of the underlying structures and principles. We will provide many examples and ways to construct new order reversing quasi-involutions from given ones. This talk is based on joint work with Shiri Artstein-Avidan and Shay Sadovsky.

This seminar will be held both online and in person:

• Room: MC 5479
• Zoom link: https://uwaterloo.zoom.us/j/94186354814?pwd=NGpLM3B4eWNZckd1aTROcmRreW96QT09

Tianyi Zheng, UC San Diego

"Random walks on self-similar groups and conformal dimension"

Conformal dimension was introduced in the late 1980s by P. Pansu; it is a natural invariant in the study of the geometry of hyperbolic spaces and their boundaries. In this talk we will discuss how conformal geometry can be used to study random walks on iterated monodromy groups, in particular, random walk entropy bounds when the limit set has Ahlfors-regular conformal dimension strictly less than 2. Based on joint work with N. Matte Bon and V. Nekrashevych.

MC 5501

Jakub Krásenský, Czech Technical University in Prague

"Criterion sets for quadratic forms over number fields"

By the celebrated 15 theorem of Conway and Schneeberger, a classical positive definite quadratic form over Z is universal if it represents each element of {1,2,3,5,6,7,10,14,15}. Moreover, this is the minimal set with this property. In 2005, B.M. Kim, M.-H. Kim and B.-K. Oh showed that such a finite criterion set exists in a much general setting, but the uniqueness of the criterion set is lost. Since then, the question of uniqueness for particular situations has been studied by several authors.

We will discuss the analogous questions for totally positive definite quadratic forms over totally real number fields. Here again, the existence of criterion sets for universality is known, and Lee determined the set for Q(sqrt5). We will show the uniqueness and a strong connection with indecomposable integers. A part of our uniqueness result is (to our best knowledge) new even over Z. This is joint work with G. Romeo and V. Kala.

Zoom link: https://uwaterloo.zoom.us/j/98937322498?pwd=a3RpZUhxTkd6LzFXTmcwdTBCMWs0QT09

Rachael Alvir, Department of Pure Mathematics, University of Waterloo

"Computable Structure Theory X"

We will conclude our series of talks on computable structure theory.

MC 5479

Gian Sanjaya, Department of Pure Mathematics, University of Waterloo

"Arithmetic Schemes"

We now look at examples of arithmetic schemes.

MC 5417